E7Lecture14Summary - E7 Lecture 14 Generalized Linear Least...

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E7 Lecture 14: Generalized Linear Least Squares Regression Tad Patzek, Civil & Environmental Engineering, U.C. Berkeley March 19, 2008
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Subjects Already Covered. . . In Lecture 11, we have learned about Mean, variance, standard deviation, covariance Example, trends of cigarette data Least squares regression to a straight line Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.1/28
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Subjects Covered in Lecture 14 Today we will learn about Generalized Linear Least Squares regression to a polynomial The related MATLAB files and Lecture 14 have been posted on bspace Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.2/28
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Population sample. . . Let y 1 , y 2 , y 3 , . . . , y N represent a random sample of size N from any population Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.3/28
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Sample mean μ . . . Estimate of ˜ μ = μ = = y 1 + y 2 + ··· + y N N = N i =1 y i N Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.4/28
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Sample variance s 2 . . . Estimate of σ 2 = s 2 = N i =1 ( y i - μ ) 2 N - 1 Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.5/28
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Sample standard deviation s . . . Estimate of σ = s = radicalBigg N i =1 ( y i - μ ) 2 N - 1 Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.6/28
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Fitting data to straight line. . . We have N measurements of a response variable ( e.g. , nicotine content), { y i } , at discrete values of an explanatory variable ( e.g. , tar content or time), { x i } , where N is very large Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.7/28
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Fitting data to straight line. . . At first, we want to consider the simplest possible model of our data, a straight line: y ( x ) = a 0 + a 1 x = y ( x ; a 0 , a 1 ) This problem is called linear regression Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.8/28
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Fitting data to straight line We assume that the uncertainty (noise), σ i , associated with each measurement y i is known, and that we know exactly each value of x i Prof. T.W. Patzek’s E7 Lecture 14: . . . – p.9/28
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Example. . .
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